Definition 4.1.3 (Deligne canonical extension [1]}) 
With notation as in definition:at-infinity,
let \((E, \nabla )\)  be a flat holomorphic vector bundle on \(X\) . The Deligne canonical extension
\((\overline{E}, \overline{\nabla }: \overline{E}\rightarrow \overline{E}\otimes \Omega ^1_{\overline{X}}(\log Z))\)  of \((E, \nabla )\)  to \(\overline{X}\)  is the
unique flat vector bundle on \(\overline{X}\)  with regular singularities
along \(Z\) , equipped with an isomorphism \((\overline{E},\overline{\nabla })|_X\overset{\sim }{\rightarrow }(E, \nabla )\) , characterized by the
property that all eigenvalues of its residues
along components of \(Z\) 
have real parts lying in \([0,1)\) .

